1. 2 Link Planar Manipulator  2  1 a 1 a 2 O 2 O 1 O 0 x 1 x 0 x 2 y 1 y 2 y 0 Frame 0 – ground reference Frame 1 – link 1, distal end Frame 2 – link 2, distal end Length of Link 1 = a 1 Length of Link 2 = a 2 Note: coordinate systems are consistent with the Denavit-Hartenburg system

3. Homogeneous Transformations in a plane O 1 O 0 p 0 p 1  0 d 01 y 0 x 0 x 1 y 1 P

5. Frame 1 is displaced from Frame 0 by rotation of 30 degrees and a translation of (1,1). Frame 2 is displaced from Frame 1 by a rotation of 60 degrees and translation of (1/2,  3/2) O 1 O 0 p 2 30 d 01 y 0 x 0 x 1 y 1 1 1 d 12 p 1 p 0 y 2 x 2 60 1/2 Sqrt(3)/2 1 1

6. O 1 O 0 p 2 30 d 01 y 0 x 0 x 1 y 1 1 1 d 12 p 1 p 0 y 2 x 2 60 1/2 Sqrt(3)/2 1 1

8. P1 P2

9. Operators Note: in composing homogeneous transformations Translation matrix, then Rotation Matrix